PreAP FDN PARALLEL LINES & 2.2 ANGLES IN PARALLEL LINES

Transcription

1 PreAP FDN PARALLEL LINES & 2.2 ANGLES IN PARALLEL LINES Congruent Vs Equal: Congruence is a relationship of shapes and sizes, such as segments, triangles, and geometrical figures, while equality is a relationship of sizes, such as lengths, widths, and heights. Congruence deals with objects while equality deals with numbers. You don t say that two shapes are equal or two numbers are congruent. Concepts: #17 Supplementary Angles: Complimentary Angles: DO: Draw 2 parallel lines. How do you know they are parallel? - Draw a line crossing both parallel lines ( transversal) - Label all angles What angles are the same? What angles are supplementary? - Make some conjectures about these angles Definitions: 1) Transversal: 2) Vertically Opposite Angles THEOREM A: In a pair of intersecting lines the vertically opposite angles are and will have measures Foundations 20(Ms. Carignan) FM20.4 Parallel Lines & Polygons (Ch 2) Page 1

2 3) Interior Angles: THEOREM B: When a transversal intersects two parallel lines, Alternate Interior angles are and will have measures 4) Exterior Angles: THEOREM C: When a transversal intersects two parallel lines, Alternate Exterior angles are and will have 5) Corresponding Angles: THEOREM D: When a transversal intersects two parallel lines, Corresponding angles are and will have Converse of Theorem D: o When a transversal intersects a pair of lines creating equal corresponding angles, the two lines are parallel. o When a transversal intersects a pair of nonparallel lines the corresponding angles are not equal ( and vice versa) Foundations 20(Ms. Carignan) FM20.4 Parallel Lines & Polygons (Ch 2) Page 2

3 6) Same-Side Interior Angles THEOREM D: When a transversal intersects two parallel lines, the Same Side Interior Angles are meaning that they add to EXAMPLE #1: : Based on the given info are lines x and y parallel? Why or why not? EXAMPLE #2: : Find the value of x that makes j // k a) b) EXAMPLE #3: And explain your reasoning. Foundations 20(Ms. Carignan) FM20.4 Parallel Lines & Polygons (Ch 2) Page 3

4 (Concept #17) FA: P72 #5 P 78 #1-4, 20 PLUS SOME of the following: Foundations 20(Ms. Carignan) FM20.4 Parallel Lines & Polygons (Ch 2) Page 4

5 PreAP FDN 20 REFERENCE SHEET: THEOREMS & POSTULATES FOR PROOF Concepts: #18 & 21 Foundations 20(Ms. Carignan) FM20.4 Parallel Lines & Polygons (Ch 2) Page 5

6 Foundations 20(Ms. Carignan) FM20.4 Parallel Lines & Polygons (Ch 2) Page 6

7 Foundations 20(Ms. Carignan) FM20.4 Parallel Lines & Polygons (Ch 2) Page 7

8 PreAP FDN & 2.2 ANGLE & PARALLEL LINE PROOFS TWO COLUMN PROOFS REVIEW Two column proofs are similar to proving that someone in court is telling the truth. A series of STATEMENTS are made that will go towards proving that either the mathematical concept or the person on trial is telling the truth. Each statement must be backed up with a REASON in the same way that each statement in court must have a corroborating witness. o Even the obvious facts must be presented in a statement and backed up with a reason. The last statement in a proof is always what is being proven. Concepts: #18 If there is not a reason given to back up a statement, the truth of the statement is left in doubt. If the statements and reasons listed lead to a different conclusion than the one attempting to be proven, the original assumption of truth cannot be determined. Every mathematical rule and theorem and postulate that is learned in school, has been proven to be true. Many of these were proven to be true using the two column proof format. Foundations 20(Ms. Carignan) FM20.4 Parallel Lines & Polygons (Ch 2) Page 8

9 EXAMPLE #1: Use a two column proof to deductively prove that alternate interior angles of parallel lines are equal. STATEMENTS REASONS EXAMPLE #2: Use a two column proof to deductively prove that same side interior angles of parallel lines are supplementary. STATEMENTS REASONS Foundations 20(Ms. Carignan) FM20.4 Parallel Lines & Polygons (Ch 2) Page 9

10 EXAMPLE #3: STATEMENTS REASONS EXAMPLE #4: Use a two column proof to deductively prove that ST = TR STATEMENTS REASONS Foundations 20(Ms. Carignan) FM20.4 Parallel Lines & Polygons (Ch 2) Page 10

11 EXAMPLE #5: EXAMPLE #6: Prove that AB // CD. STATEMENTS REASONS (Concept #18) FA: P79 #8,10, 12, 15, 16, 18 P 78 #1-4, 20 MLA: P79 #9, 17, 19, Helpful hints for the assignment: An isosceles triangle has two sides of equal length and two equal angles. If a line bisects an angles it divides the angle into two equal angles. Foundations 20(Ms. Carignan) FM20.4 Parallel Lines & Polygons (Ch 2) Page 11

12 PreAP FDN ANGLE PROPERTIES IN TRIANGLES & PROOFS Recall: All interior angles of a triangle add up to. EXAMPLE #1: prove, deductively, that the sum of the measures of the interior angles of any triangle is. Statement Reason Concepts: #17, 18, 20 Note: Exterior angles are formed by extending a side of a polygon. For example, extend one side of this triangle to make an exterior angle: EXAMPLE #2: In the diagram, angle MTH is an exterior angle of ΔMAT. Determine the measures of the unknown angles in ΔMAT. What relationship do you notice about angle AMT, angle MAT and the exterior angle MTH? Foundations 20(Ms. Carignan) FM20.4 Parallel Lines & Polygons (Ch 2) Page 12

13 EXAMPLE #3: Prove deductively using a two column proof that an exterior angle of a triangle is equal to the sum of the two non- adjacent sides. Statement Justification EXAMPLE #4: In EFG, GI bisects a) If E y, prove GI // EF. FGH b) Prove EF // GI, if F z EXAMPLE #4: GIVEN: BAC is isosceles m 1 40 m F PROVE: BC DE G Foundations 20(Ms. Carignan) FM20.4 Parallel Lines & Polygons (Ch 2) Page 13

14 (Concept #17, 18, 20) FA: P90 #2, 3, 5, 7, 9, 19-12, 14, 15 PLUS the following proofs MLA: P90 #16, GIVEN: PROVE: a 25 STATEMENTS REASONS Given 2. Given 3. Given 4. c Corresponding s of lines are 6. a Foundations 20(Ms. Carignan) FM20.4 Parallel Lines & Polygons (Ch 2) Page 14

15 PreAP FDN ANGLE PROPERTIES IN POLYGONS Concepts: #19, 20 What is the interior angle sum of any triangle? (we proved this last section) What will the 4 interior angles of any quadrilateral always add to? The 5 interior angles of a pentagon? Let s investigate: Polygon # of Sides # of Triangles Sum of Interior Angle Measures Triangle Quadrilateral Pentagon Hexagon Heptagon Octagon Foundations 20(Ms. Carignan) FM20.4 Parallel Lines & Polygons (Ch 2) Page 15

16 How can we find the sum of the interior angles based on the number of sides of a polygon has? Sum of the Interior Angles of a Convex Polygon: Given that n = the number of sides Interior Angle Sum = Measure of each Interior Angle of a REGULAR Convex Polygon: Given that n = the number of sides of equal length Each Interior Angle = EXAMPLE #1: a) What would be the sum of the measures of the interior angles of a regular dodecagon (12-sides)? b) Determine the measure of each interior angle of a regular dodecagon? Sum of the EXTERIOR ANGLES of a Convex Polygon: Given that n = the number of sides EXTERIOR Angle Sum = Measure of each EXTERIOR Angle of a REGULAR Convex Polygon: Given that n = the number of sides of equal length Each Interior Angle = EXAMPLE #2: : Deductively prove that the sum of the exterior angles of any polygon will be Foundations 20(Ms. Carignan) FM20.4 Parallel Lines & Polygons (Ch 2) Page 16

17 EXAMPLE #3: : Bob is tiling his floor. He uses regular hexagons and regular triangles. The side length of a triangle is equal to the side length of a hexagonal tile. Can he tile the floor without leaving any gaps between tiles?( Concept 20) EXAMPLE #3: Kieran drew a 14 sided convex polygon. One of the interior angle measures 155 0, Is it a regular polygon? (Concept #19, 20) FA: P99 #1, 2, 37, 10, 16, 18 MLA: P99 #13 ULA: P99 #20, 21 Foundations 20(Ms. Carignan) FM20.4 Parallel Lines & Polygons (Ch 2) Page 17

18 PreAP FDN CONGRUENT TRIANGLES (Not in Textbook) Two triangles are considered to be CONGRUENT ( ) when the following is true: Concepts: #21 AND This means that (we could also write this as or or or or ) NOTE: The order of the letters in the first triangle must correspond to the correct order in the second triangle The following are congruent between the two triangles: ANGLES: and SIDES: THERE ARE FIVE WAYS TO DETERMINE IF TWO TRIANGLES ARE CONGRUENT: Foundations 20(Ms. Carignan) FM20.4 Parallel Lines & Polygons (Ch 2) Page 18

19 Note: Angle, Side, Side is not enough information to conclude that the triangles are congruent. As two different triangles can be made is an angle and is opposing sides are congruent. EXAMPLE #1: Foundations 20(Ms. Carignan) FM20.4 Parallel Lines & Polygons (Ch 2) Page 19

20 EXAMPLE #2: Determine if the following sets of triangles would be congruent using the above five reasons: SSS, ASA, SAS, AAS or HL. State the triangle congruency if there is one. a) b) by by c) d) by by e) f) by by g) h) by by Foundations 20(Ms. Carignan) FM20.4 Parallel Lines & Polygons (Ch 2) Page 20

21 (Concept #21) 1. a) b) 2. Determine if the following sets of triangles would be congruent using the above five reasons: SSS, ASA, SAS, AAS or HL. State the triangle congruency if there is one. a) b) c) d) e) f) g) h) i) j) k) Foundations 20(Ms. Carignan) FM20.4 Parallel Lines & Polygons (Ch 2) Page 21

22 3. a) b) c) d) Answers: 1a) b) a) A D, B E, C F T I IG CA ATC BGI b) AB DE, AC DF, BC EF c) NO d) Yes 2) a) SSS; ABC CDA b) SAS; MKL BCA c) Not enough info to determine if they are congruent d) AAS; WPB GDH e) SSS; ABC NMO f) Not enough info g) SAS; ABC YXZ h) HL; EQD DHE i) ASA; ABC ADC j) HL; AUE TWK k) SAS; ACB ECD 3.) a) N b) CE c) D d) R Foundations 20(Ms. Carignan) FM20.4 Parallel Lines & Polygons (Ch 2) Page 22

23 PreAP FDN PROVING TRIANGLES CONGRUENT (Not in Textbook) Refer to your reference sheet on reasons to use in a two column proof Concept: #21 EXAMPLE #1: GIVEN: O STATEMENTS REASONS L E V PROVE: LOE VOE EXAMPLE #2: 30 STATEMENTS REASONS Foundations 20(Ms. Carignan) FM20.4 Parallel Lines & Polygons (Ch 2) Page 23

24 EXAMPLE #3: : STATEMENTS REASONS EXAMPLE #4: GIVEN: STATEMENTS REASONS PROVE: XY VW Foundations 20(Ms. Carignan) FM20.4 Parallel Lines & Polygons (Ch 2) Page 24

25 (Concept #21) Foundations 20(Ms. Carignan) FM20.4 Parallel Lines & Polygons (Ch 2) Page 25

26 Use a two column proof for the following Foundations 20(Ms. Carignan) FM20.4 Parallel Lines & Polygons (Ch 2) Page 26

27 Extra QUESTIONS IF NEEDED Foundations 20(Ms. Carignan) FM20.4 Parallel Lines & Polygons (Ch 2) Page 27

28 Foundations 20(Ms. Carignan) FM20.4 Parallel Lines & Polygons (Ch 2) Page 28